Accepted Manuscript
Research papersThe added value of stochastic spatial disaggregation for short-term rainfall fore-casts currently available in Canada
Patrick Gagnon, Alain N. Rousseau, Dominique Charron, Vincent Fortin, René Audet
PII: S0022-1694(17)30549-8
DOI: http://dx.doi.org/10.1016/j.jhydrol.2017.08.023
Reference: HYDROL 22186
To appear in: Journal of Hydrology Received Date: 4 January 2017 Revised Date: 10 August 2017 Accepted Date: 14 August 2017
Please cite this article as: Gagnon, P., Rousseau, A.N., Charron, D., Fortin, V., Audet, R., The added value of stochastic spatial disaggregation for short-term rainfall forecasts currently available in Canada, Journal of Hydrology (2017), doi: http://dx.doi.org/10.1016/j.jhydrol.2017.08.023
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1
The added value of stochastic spatial disaggregation for short-term rainfall forecasts 1
currently available in Canada 2
Patrick Gagnona1, Alain N. Rousseaub*, Dominique Charronc, Vincent Fortind, René 3
Audete 4
5
a
Agriculture and Agri-Food Canada, 2560 Hochelaga blvd, Québec city, Qc, Canada, 6
G1V 2J3. [email protected] 7
b
Institut National de la Recherche Scientifique, Centre Eau Terre Environnement, 490 rue 8
de la Couronne, Québec city, Qc, Canada, G1K 9A9. [email protected] 9
(*Corresponding author) 10
c
Institut National de la Recherche Scientifique, Centre Eau Terre Environnement, 490 rue 11
de la Couronne, Québec city, Qc, Canada, G1K 9A9. [email protected] 12
d
Environment and Climate Change Canada, 2121 route Transcanadienne, Dorval, Qc, 13
Canada, H9P 1J3. [email protected] 14
e
Agriculture and Agri-Food Canada, 2560 Hochelaga blvd, Québec city, Qc, Canada, 15 G1V 2J3. [email protected] 16 17 1
Current address: Centre de développement du porc du Québec inc. Place de la Cité, Belle Cour tower, 2590 Laurier blvd, office 450, Québec city, Québec, G1V 4M6, [email protected]
2
Abstract
18
Several businesses and industries rely on rainfall forecasts to support their day-to-day 19
operations. To deal with the uncertainty associated with rainfall forecast, some 20
meteorological organisations have developed products, such as ensemble forecasts. 21
However, due to the intensive computational requirements of ensemble forecasts, the 22
spatial resolution remains coarse. For example, Environment and Climate Change 23
Canada’s (ECCC) Global Ensemble Prediction System (GEPS) data is freely available on 24
a 1-degree grid (about 100 km), while those of the so-called High Resolution 25
Deterministic Prediction System (HRDPS) are available on a 2.5-km grid (about 40 times 26
finer). Potential users are then left with the option of using either a high-resolution 27
rainfall forecast without uncertainty estimation and/or an ensemble with a spectrum of 28
plausible rainfall values, but at a coarser spatial scale. The objective of this study was to 29
evaluate the added value of coupling the Gibbs Sampling Disaggregation Model (GSDM) 30
with ECCC products to provide accurate, precise and consistent rainfall estimates at a 31
fine spatial resolution (10-km) within a forecast framework (6-hours). For 30, 6-h, 32
rainfall events occurring within a 40,000-km2 area (Québec, Canada), results show that, 33
using 100-km aggregated reference rainfall depths as input, statistics of the rainfall fields 34
generated by GSDM were close to those of the 10-km reference field. However, in 35
forecast mode, GSDM outcomes inherit of the ECCC forecast biases, resulting in a poor 36
performance when GEPS data were used as input, mainly due to the inherent rainfall 37
depth distribution of the latter product. Better performance was achieved when the 38
Regional Deterministic Prediction System (RDPS), available on a 10-km grid and 39
aggregated at 100-km, was used as input to GSDM. Nevertheless, most of the analyzed 40
3
ensemble forecasts were weakly consistent. Some areas of improvement are identified 41
herein. 42
Highlights
43• GSDM applied on reference data generates accurate and precise rainfall fields 44
• GSDM inherits the forecast bias 45
• Global ensemble forecasts (GEPS) are not consistent, with or without GSDM 46
• GSDM performance was better with regional deterministic forecasts (RDPS) 47
• Neighboring pixels should be considered when producing high-resolution 48
ensembles 49
Keywords
50Gibbs Sampling Disaggregation Model (GSDM), Canadian Precipitation Analysis 51
(CaPA), Global Ensemble Prediction System (GEPS), Regional Deterministic Prediction 52
System (RDPS), ensemble, high-resolution rainfall. 53
4
1 Introduction
55
Numerous businesses and industries need consistent, precise and accurate, high-56
resolution, short-term rainfall forecasts. For example, in urban areas, high-resolution 57
rainfall forecasts are of interest for stormwater management (e.g., Gaborit et al., 2014). 58
They are also important in rural areas where agricultural activities (e.g., fungicide 59
applications, some herbicides applications, hay harvesting, manure application, irrigation 60
management), that are affected by local rainfall depth, require up-to-the-hour information 61
(e.g., Cai et al., 2007; Silva et al., 2010; Cai et al., 2011; Bendre et al., 2015). 62
Thanks to advances in computational resources, understanding and parameterization of 63
key physical processes, increased access to satellite data, and data assimilation 64
techniques, meteorological models have made tremendous strides in the last decades 65
(Bauer et al., 2015). One of the most spectacular changes that has occurred is an 66
impressive increase in horizontal resolution. For example, the horizontal resolution of 67
Environment and Climate Change Canada’s (ECCC) global deterministic prediction 68
system has improved from approximately 150-km in 1990 to 25-km in 2013 (changes to 69
ECCC’s operational model are documented at:
70
http://collaboration.cmc.ec.gc.ca/cmc/cmoi/product_guide/docs/changes_e.html). 71
Throughout Canada, deterministic forecasts are routinely issued on a grid having a 72
horizontal resolution of 2.5 km. The accuracy of the forecasts has improved accordingly. 73
For example, based on the root mean-squared-error (RMSE) of the 850 hPa temperature 74
forecasts issued by ECCC over North America, a 3-day forecast issued in 2015 was about 75
as accurate as a 1-day forecast issued in 1995, and a 5-day forecast issued in 2015 was 76
about as accurate as a 3-day forecast issued in 1995. This corresponds to a gain of 77
5
approximately one day of lead time per decade, which is consistent with improvements 78
reported by Bauer et al. (2015) for the European Centre for Medium-Range Weather 79
Forecasts. 80
Despite significant advances (e.g., COSMO forecast system; Baldauf et al., 2011), it 81
remains difficult to get reliable rainfall forecasts for fine spatiotemporal scales. To 82
circumvent this issue, meteorological organisations are developing ensemble products 83
that provide several forecasts for a given timeframe; providing a spectrum of rainfall 84
depths associated with model uncertainty. However, because of the ensuing 85
computational requirements, the spatial resolution is generally coarser than that of a 86
deterministic run. Furthermore, outputs are not always made available on the original 87
model grid due to disk space constraints. For example, the Canadian ensemble forecasts 88
from the Global Ensemble Prediction System (GEPS) of ECCC, which are issued on a 89
50-km grid, are freely available on a 1-degree grid (about 100 km), while the spatial 90
resolution of their High Resolution Deterministic Prediction System (HRDPS) is about 91
40 times finer (2.5 km at 60°N). Thus, there is still a wide gap between available 92
forecasts and stakeholder requirements, namely: (i) rainfall estimates close to actual, 93
local-scale values, (ii) information about the uncertainty of local estimates, and for some 94
applications, (iii) available data for short-term decisions. 95
Meanwhile, rainfall and weather generators can both produce fine-scale rainfall fields 96
from coarse meteorological and/or climate simulations (e.g., Paschalis et al., 2013; Peleg 97
and Morin, 2014; Niemi et al., 2016). However, the goal of most applications is to 98
generate scenarios for long-term predictions. Alternatively, disaggregation models which 99
6
can generate rapidly several fine-scale rainfall fields from one coarse scale field represent 100
a promising avenue for short-term forecasts. 101
During the past decades, several studies have focused on spatial distribution of rainfall at 102
fine scale (Gupta and Waymire, 1993; Hubert et al., 1993; Kumar and Foufoula-103
Georgiou, 1993a,b; Marsan et al., 1996; Olson and Niemczynowicz, 1996; among 104
others). An approach commonly used by disaggregation models is to divide in cascade 105
each grid cell in 2x2 pixels, which are then divided in 2x2 sub-pixels, so on so forth (e.g., 106
Over and Gupta, 1996; Perica and Foufoula-Georgiou, 1996; Deidda, 2000; Harris and 107
Foufoula-Georgiou, 2001; Badas et al., 2006; Deidda et al., 2006a,b; Gaborit et al., 2014; 108
among others). These models are conceptually simple, but may lead to unrealistic rainfall 109
fields with visible discontinuities, due to the discretization of the space (Lovejoy and 110
Schertzer, 2010a,b). Gagnon et al. (2012) proposed a stochastic disaggregation model, 111
hereafter referred to as the Gibbs Sampling Disaggregation Model (GSDM), which does 112
not produce discontinuities, even for adjacent pixels from two different grid cells (Figure 113
1). The model was adapted for orographic rainfall (Gagnon et al., 2013) and used to 114
evaluate the impact of climate change on extreme rainfall events over a small watershed 115
(Gagnon and Rousseau, 2014). Nevertheless, the model has never been applied for short-116
term meteorological forecasts. 117
The general objective of this study was to evaluate the capability of GSDM coupled with 118
ECCC products to provide accurate, precise and consistent rainfall estimates within a 119
short time frame. The specific objectives were to: 120
7
(i) Evaluate accuracy, precision and consistency of GSDM when applied on 121
aggregated reference rainfall fields; 122
(ii) Compare two freely-available ECCC rainfall forecasts, a deterministic and an 123
ensemble forecast, to a reference rainfall product (namely CaPA, cf Section 2.2); 124
(iii)Determine the added value of coupling GSDM to these ECCC forecasts; 125
(iv) Identify possible modifications to GSDM potentially leading to improved 126
forecasts. 127
This study focused on 30, 6-h, rainfall events that occurred between July and November 128
2015 over a 40,000-km2 area on the south shore of the St. Lawrence River, Québec, 129
Canada. 130
2 Materials and Methods
1312.1 Study Area 132
The region of interest consists of an area approximately 200 x 200 km2, from 45 to 47° N 133
and from 71 to 73° W, covering the watershed of the Bécancour River on the south shore 134
of the St. Lawrence River in Québec, Canada (Figure 2). Primarily located in the St. 135
Lawrence Lowlands, only the upstream southeastern portion is in the Appalachian 136
Mountains. The climate is continental humid (Dfb under Köppen classification) with the 137
highest precipitation during the months of July, August and September. Around 40% of 138
the area, mainly in the Lowland sector, is occupied by agricultural activities that would 139
benefit from better local-scale rainfall forecasts. With respect to GDSM data 140
requirements, the modeled area spans from 44 to 48° N and from 70 to 74° W (study area 141
± 1° in each direction). In this manuscript, a grid cell refers to the spatial unit of a grid 142
8
with 1° (about 100-km) resolution and pixel refers to a spatial unit of a 10-km grid (see 143
Figure 2). 144
2.2 Environment and Climate Change Canada products 145
Three different ECCC products, all freely available on the web 146
(https://weather.gc.ca/grib/index_e.html), were used. First, the Canadian Precipitation 147
Analysis (CaPA; Fortin et al., 2015) was identified as the reference (pseudo-observed) 148
precipitation. CaPA uses short-term forecasts as a background field and assimilates data 149
from various sources (stations, radars, satellites). The background field is modified by 150
spatial interpolation (kriging) of the difference between the forecast and the observations. 151
The dataset has a spatial resolution of 10 km x 10 km on a polar stereographic grid 152
covering North America. For this study, the computational domain consisted of a 40x40-153
pixel grid (Figure 2) overlaying the area of interest (44-48°N and 70-74°W), oriented 154
east-west, by taking the closest pixel from the original grid. 155
Second, the Regional Deterministic Prediction System (RDPS; Caron et al., 2015) 156
forecasts on the same grid as that of CaPA, similarly transposed on the 40x40-pixel grid 157
oriented east-west. It is important to note that RDPS provides the background field to 158
CaPA, the reference precipitation, and thus it could be an advantage for RDPS compared 159
to another product. That being said, the dense observation network and the radar in the 160
region analyzed provided sufficient data limiting the contribution of RDPS in CaPA. It is 161
thus assumed that the advantage of RDPS, if any, is negligible for the studied region. 162
The third and final product was the Global Ensemble Prediction System (GEPS; Charron 163
et al., 2010; Houtemaker et al., 2014) which produces 21 forecasts on a 1° x 1° (about 164
100 km x 100 km) grid. The 21 members are formed by one control member and 20 165
9
perturbed members, having different physical parameterization, data assimilation cycles 166
and initial observed conditions. 167
In addition to rainfall depth, three input atmospheric variables required by GSDM 168
(Section 2.3) were obtained from RDPS and GEPS forecasts: (i) convective available 169
potential energy (CAPE), (ii) wind speed and (iii) wind direction at the 700-hPa pressure 170
level. 171
CaPA and GEPS data are available at a 6-h time step, while RDPS data at a 3-h time step, 172
but aggregated at 6-h time step (sum for rainfall depth, average for wind speed and 173
CAPE). For GEPS, since simulations are launched twice a day, the required data covered 174
two forecast periods: (i) the first 6 hours and (ii) from 6 to 12 hours following the start of 175
the simulations. The same strategy is used for the RDPS, although RDPS forecasts are 176
available four times per day. 177
Two individual periods were analyzed for GSDM calibration and evaluation of rainfall 178
products. The calibration period covered May through October 2014 as well as May and 179
June 2015. Each CaPA grid point that received at least 1 mm of rain during a 6-h time 180
step was retained for calibration, for a total of 208,766 pixels. Atmospheric variables 181
(CAPE and wind at 700 hPa) were retrieved from RDPS forecasts. Evaluation of ECCC 182
products (Section 2.4) spanned from July to November 2015. A total of 30, 6-h time steps 183
with the largest mean GEPS forecasted accumulations were retained (Table 1). 184
Note that ECCC has other forecast products that could be of interest, but were not 185
included in the present study such as the High Resolution Deterministic Prediction 186
System (HRDPS; Mailhot et al., 2010), having a 2.5-km resolution. Also, the Regional 187
10
Ensemble Prediction System (REPS; Charron et al., 2013) having grid spacing of 15-km 188
and a lead time of 72-h, with two integrations per day and 21 ensemble members, was not 189
included since the data is not freely available. Moreover, it could be difficult to retrieve 190
in a timely fashion for actual short-term forecasts. 191
2.3 Gibbs Sampling Disaggregation Model (GSDM) 192
The model assumed that Ri,j, the rainfall depth at a given 10-km pixel (i,j) for a given 6-h
193
period, is a random variable with expected value µ and standard deviation σ given by 194
(Gagnon, 2012; Gagnon and Rousseau, 2014): 195
(
)
(
)(
)
(
(
)
)(
)
[
−]
− − − + − − + + − + + =A d A A| A/ A\ vV W A/ A\ W A| A 90 2 cos 45 2 cos 2 2β
β
µ
196 (1) 197(
)
2 1 0 β µ β β σ = + C (2) 198where A is the mean rainfall depth of the eight surrounding pixels and the other A’s are 199
mean rainfall depths in the four directions, namely / 1, 1 1, 1 2 i j i j R R A = − − + + + , 200 1, 1, | 2 i j i j R R A = − + + , 2 1 , 1 1 , 1 \ − + + − + = Ri j Ri j A , and , 1 , 1 2 i j i j R R A− = − + + , V is the 700-hPa 201
wind speed (m/s), W is the 700-hPa wind direction (degree), C stands for CAPE (J/kg), 202
and the five calibration parameters are: βd (dimensionless), βv (s/m), β0 (mm), β1 (mm
203
kg/J), and β2 (dimensionless). As in many rainfall models (e.g., Over and Gupta, 1996;
204
Fiorucci et al., 2001; Forman et al., 2008; Groppelli et al., 2011), a lognormal 205
distribution is assumed for Ri,j. In these equations, it is assumed that strong 700-hPa
11
winds might lead to anisotropy and high CAPE values increase spatial variability (i.e., 207
decrease the influence of the neighboring pixels). 208
The model can disaggregate at any spatial resolution, but it is recommended to target a 209
resolution at which rainfall depths are available for calibration. In this study, 10-km 6-h 210
rainfall depths from 208,766 pixels from CaPA analyses were used for calibration 211
(Section 2.2). The estimated values for parameters βd and βv of Equation (1) minimizes
212
the sum of the squared differences between observed rainfall depths and expected rainfall 213
depths calculated using Equation (1) for all pixels used for calibration (Gagnon, 2012; 214
Gagnon et al., 2012). Then, groups were created from all calibration pixels; all pixels 215
within a group had similar expected rainfall depths and CAPE values. For each group, 216
mean expected rainfall depth and mean CAPE value were calculated. The estimated 217
values for parameters β0, β1, and β2 of Equation (2) minimizes the sum of the squared
218
differences between the observed 99.9% quantile of rainfall depths in each group and the 219
99.9% quantile calculated using the lognormal distribution with mean expected rainfall 220
depth for each group and standard deviation given by Equation (2) (with mean CAPE 221
value for each group and calibration parameters). Fitting the 99.9% quantile was done to 222
attenuate the underdispersion of the lognormal distribution for rainfall depth estimation 223
(Gagnon et al., 2012; Gagnon and Rousseau, 2014). 224
Equations (1) and (2) allow rainfall depths to be generated on a 10-km pixel when 225
neighboring depths are known. However, in practice, only coarse scale (100 km in this 226
study) data is available as input to the disaggregation model. An algorithm based on the 227
Gibbs sampling theory (Geman and Geman, 1984; Roberts and Smith, 1994) was 228
developed to circumvent this issue. First, as initial conditions, the rainfall depth for each 229
12
10-km pixel was set to the rainfall depth of the 100-km grid cell covering the pixel. Then, 230
new rainfall depths were generated from the lognormal distribution using Equations (1) 231
and (2) for each 10-km pixel, one at the time. An iteration is completed when all pixels 232
have been updated once. After each iteration, a multiplicative factor (generally close to 1) 233
was applied to ensure that the model preserved the exact rainfall depth of each 100-km 234
grid cell used as input. Based on the Gibbs sampling theory (Geman and Geman, 1984; 235
Roberts and Smith, 1994), the rainfall depth on each 10-km pixel is approximately 236
distributed using the lognormal distribution along with Equations (1) and (2) after a 237
sufficient number of iterations. In this study, 300 iterations were performed before 238
retaining the first disaggregated rainfall field, referred to as the first realization. Since 239
fields from consecutive iterations are autocorrelated, subsequent realizations were 240
separated by 100 iterations. The model is not explicitly made to generate spatial rainfall 241
intermittency (i.e., pixels without rain), but this can be achieved by setting to 0 those 242
rainfall depths below a given threshold (0.1 mm in the present work). A detailed 243
description of the algorithm is provided in Gagnon (2012), Gagnon et al. (2012) and 244
Gagnon and Rousseau (2014). 245
2.4 Products analyzed 246
For each one of the 30, 6-h, rainfall events retained (Section 2.2), a total of 13 different 247
rainfall products were analyzed (Table 2). These products were referenced with respect to 248
the ECCC product used (“C” for CaPA analyses, “R” for RDPS forecasts and “G” for 249
GEPS forecasts), whether aggregation and/or disaggregation was performed (“d” for 10-250
km disaggregated product, “a” for 100-km aggregated product without disaggregation), 251
and whether neighboring pixels were used to create an ensemble (“n” if so). 252
13
The original 10-km CaPA rainfall (C) became the reference (true value) for the 12 other 253
series. GSDM was first applied with aggregated CaPA data (Ca) as input. The ensuing 254
product (Cd) compared to the reference (C) evaluates the performance of the 255
disaggregation model. Comparison of the outcomes of Cd with Ca was used to assess the 256
added value of GSDM over a low-resolution product. 257
One of the strength of GSDM is that it can generate realistic spatial patterns, even with 258
100-km grid cells used as input. However, it cannot always be right at the exact location 259
(10-km pixel) since no fine scale information is used as input. Thus, it was decided to 260
evaluate another disaggregated ensemble product, which instead of considering only the 261
rainfall depths generated at the target pixel (as for Cd), it also includes the rainfall depths 262
in the neighboring area (Cdn; 100 model realisations x 121 pixels [+/- 5 pixels in each 263
direction] = 12,100 rainfall depths per 10-km pixel per 6-h event). The ensembles Cd and 264
Cdn were also compared with the ensemble formed by the CaPA rainfall depths in 120 265
neighboring pixels (+/- 5 pixels in each direction - the target pixel; Cn, Table 2). This 266
latter ensemble was used to evaluate whether neighboring pixels could actually be used 267
for suitable estimation of the rainfall depth of the target pixel in this area. 268
Eight forecast products were compared. For both RDPS and GEPS forecasts, analyses 269
were performed on rainfall depths from raw products (referred to as R and G, 270
respectively), in order to evaluate the actual forecasts available for an end user. Then, 271
disaggregation at the target pixel (Rd and Gd) and disaggregation in the neighboring area 272
(Rdn and Gdn) was performed to evaluate the added value of GSDM coupled with the 273
forecasts. For RDPS, it required data aggregation (Ra) prior disaggregation. For a given 274
pixel, RDPS outcomes in the neighboring area (+/- 5 pixels in each direction; Rn) were 275
14
also analyzed. The interest in Rn lies in the construction of an ensemble from a 276
deterministic forecast requiring no additional computational time, contrary to 277
disaggregation. 278
2.5 Performance evaluation 279
The performance of a product was assumed to vary depending on the reference rainfall 280
depth (large rainfall depths being more difficult to correctly place spatially) and on the 281
type of events (stratiform or convective events being governed by different physical 282
processes). Seven groups, based on these two variables, were constructed (Table 3) and 283
performance metrics calculated independently for each group. 284
Three criteria were accounted for in the evaluation: accuracy, precision and, for 285
stochastic products only, consistency. Accuracy refers to bias, that is the mean difference 286
between simulated and observed values. Precision is defined in two ways. Precision of a 287
probabilistic product is related to the variability (range) of the realizations. Precision of 288
the error, for a deterministic or probabilistic product, is related to the variability of the 289
difference between the prediction and the reference. Finally, consistency is when the 290
reference value is indistinguishable from a randomly selected member of an ensemble 291
(Anderson, 1997). 292
For all products, deterministic or probabilistic, the Mean Squared Error (MSE) was 293
calculated. For a given group with n pixels (Table 3) and a given rainfall product, let x1,
294
…, xn be the reference rainfall depths (C; Table 2) and y1, …, yn be the corresponding
295
forecasted rainfall depths from the product. If the product is probabilistic, the mean 296
forecast for each pixel was calculated to allow a comparison with deterministic products. 297
15 That is, yi =
∑
= r n j r j i n y 1 ,where nr is the number of probabilistic realizations (members) and yi,j
298
is the simulated rainfall depth for jth realization at the ith pixel. The MSE was calculated 299 as follows: 300 MSE =
(
)
(
)
(
)
(
)
(
)
2 1 1 2 2 x y n x y x y n x y n i n i i i i i − + − − − = −∑
∑
= = . (3) 301The two terms on the right-hand side of the equation correspond to the variance of the 302
error (precision of the error) and the squared of the bias (accuracy), respectively. 303
For probabilistic products only, the Cumulative Rank Probability Score (CRPS; 304
Matheson and Winkler, 1976) was calculated for each ensemble product and each group 305 (Table 3) as follows: 306 CRPS =
∑ ∫
(
( )
( )
)
= ∞ −∞ = − n i t X i Y i t F t dt F n 1 2 1 (4) 307where
F
iX( )
t
andF
iY( )
t
are, for the ith pixel of the group, the empirical cumulative 308distribution functions of the reference (C) rainfall depth (= 1 if xi ≤ t ; = 0 otherwise) and
309
of the ensemble forecasted rainfall depth, respectively. The CRPS allows for the 310
evaluation of the mean accuracy of an ensemble product while also being sensitive to the 311
width (precision of the probabilistic product) of the distribution (Hersbach, 2000). 312
Consistency of probabilistic products was evaluated via rank histograms (Talagrand 313
diagrams). They are drawn for pixels with stratiform rainfall depth between 0.1 and 5 mm 314
(Group 2, Table 3), for pixels with stratiform rainfall depth larger than 10 mm (Group 4) 315
16
and for pixels with convective rainfall depth between 0.1 and 5 mm (Group 6). Rank 316
histograms are not well suited for close-to-zero rainfall depths (Groups 1 and 5). Pixels 317
with large convective rainfall depths (Group 7) were of interest, but there were not 318
enough of them for rank histograms. Group 3 (pixels with stratiform rainfall depth 319
between 5 and 10 mm) is not shown for sake of parsimony. 320
3 Results
3213.1 Deterministic metric: MSE 322
As illustrated in Equation (3), MSE was broken down in order to verify the mean (bias; 323
Figure 3) and the standard deviation of the error (Figure 4). The results regarding the 324
CaPA-derived products illustrate that GSDM (Cd) reduced bias compared to the low-325
resolution product Ca, especially for large rainfall depths (Groups 4 and 7). The accuracy 326
(Figure 3) and precision of the error (Figure 4) are also slightly higher for Cd compared 327
to Cdn and even Cn, illustrating the ability of GSDM to generate rainfall depths close to 328
the reference depth at fine scale. Note that all products underestimated the two pixel 329
groups with the largest CaPA rainfall depths. 330
For RDPS forecasts, the bias of the product evaluated with respect to the neighborhood 331
(Rn) was similar, although slightly higher, to that of the raw product (R) (Figure 3). The 332
standard deviation of the error was lower for Rn than for R (Figure 4). In all likelihood, 333
the lower standard deviations for Rn were due to the difference in the MSE calculation 334
method; that is for the mean of the 121 ensemble members for Rn and for the unique 335
deterministic value for R (Table 2). Thus, the standard deviation for Rn is smoothed, but 336
the error of the product is not necessarily more precise. The bias of R was lower than that 337
17
of the aggregated product (Ra), illustrating the added value of higher spatial-resolution 338
forecasts. In all likelihood, the standard deviation of the error was smaller for Ra than for 339
R, because the former values were spatially smoothed. Biases of the disaggregated RDPS 340
products (Rd and Rdn) were similar to that of the raw product R. 341
Biases for the GEPS-derived forecasts were all very high (Figure 3). The coarse spatial 342
resolution of the raw GEPS product (G) did not lead to smaller and larger intra-tile 343
rainfall depths. The bias of G was larger than that of Ra, which has the same spatial 344
resolution (about 100 km), but built using a forecast with higher spatial resolution 345
(RDPS). Disaggregation (Gd and Gdn) did not provide a way to reduce the bias. As 346
mentioned earlier, a suitable application of GSDM requires an accurate (unbiased), low-347
resolution, rainfall depth. Obviously, this assumption was not met for GEPS 348
disaggregation. 349
3.2 Probabilistic metrics: CRPS and rank histograms 350
Ensembles produced by GSDM from aggregated CaPA analyses resulted in small CRPS 351
values (Figure 5). Again, Cd slightly outperformed Cdn and even Cn; illustrating the 352
ability of GSDM to produce accurate and precise ensemble rainfall depths at fine spatial 353
scale. However, for stratiform events, Cdn outperformed Cd for consistency on pixels 354
with rainfall depths between 0.1 and 5 mm (Figure 6) and larger than 10 mm (Figure 7). 355
This latter figure shows that Cd too often underestimated pixels with large rainfall depths 356
while Cdn did not have this issue. It suggests that GSDM may be not able to put the 357
largest rainfall depths at the exact location, but it can generate these large rainfall depths 358
in a neighboring area. For pixels with convective rainfall depths between 0.1 and 5 mm, 359
consistency of Cd was better (Figure 8). 360
18
RDPS-derived forecasts had generally smaller CRPS values than those of GEPS-derived 361
forecasts (Figure 5). For RDPS-derived forecasts, CRPS values were almost always 362
smaller for Rd (RDPS disaggregated and evaluated at the target pixel) than for forecasts 363
evaluated at the neighborhood pixels, disaggregated (Rdn) or not (Rn). However, Rd is 364
not consistent for the three groups of pixels analyzed (Figures 6-8). For most of the pixels 365
in each group, the reference value is either smaller than or equal to the first 5% or larger 366
than the last 5% of the stochastic realizations. Rd overestimated too often pixels with 367
stratiform rainfall depths between 0.1 and 5 mm (Figure 6). Rdn had better consistency 368
than Rd, but just slightly lower than Rn (Figures 6-8). That being said, Rn overestimated 369
pixels with convective rainfall depths between 0.1 and 5 mm (Figure 8). It suggests that 370
RDPS produced too smooth forecasts for convective rainfall. 371
For GEPS-derived forecasts, CRPS values for Gd were also smaller than those for Gdn 372
for stratiform rainfalls, but not for convective rainfalls. Consistency was weak for all 373
GEPS-derived forecasts (Figures 6-8), except for Gdn for pixels with stratiform rainfall 374
depths larger than 10 mm (Figure 7). 375
4 Discussion
376The above results demonstrated the capacity of GSDM to generate accurate and precise 377
ensemble rainfall depths at a local scale (10 km) for cases when spatially averaged 378
reference rainfall (Ca) was used as input. The bias and the standard deviation of the error 379
of the disaggregated product Cd always remained smaller than for the low-resolution 380
reference product Ca used as input (Figures 3 and 4). Also, CRPS values for Cd were 381
smaller than the reference ensemble Cn formed from 120 neighboring pixels (+/- 5 in 382
19
each direction) (Figure 5). However, the disaggregated product Cd was not consistent for 383
stratiform rainfall (Figures 6-7). Including the neighboring pixels in the disaggregated 384
ensemble (Cdn) mitigated the lack of consistency, especially for large stratiform rainfall 385
depths (Figure 7). For convective rainfall (Figure 8), Cd was consistent, thanks to the 386
parameterization of GSDM which adjusts the spatial variability according to CAPE. 387
For the ECCC forecast products analyzed, despite a high bias for large convective rainfall 388
depths (Figure 3), the regional 10-km resolution product (RDPS; R) outperformed the 389
global 100-km resolution ensemble product (GDPS; G) based on accuracy and precision 390
of the error criteria. However, RDPS is a deterministic product and does not provide 391
uncertainty bands for end users. This issue was circumvented by building an ensemble 392
with the RDPS rainfall depths forecasted in the neighboring area (Rn). Rn was relatively 393
consistent for small stratiform rainfall depths (Figure 6), but not as much for large 394
stratiform (Figure 7) and convective rainfall depths (Figure 8). Furthermore, GDPS 395
ensemble was clearly not consistent (Figure 6-8). 396
While the added value of GSDM applied on low-resolution reference rainfall depths is 397
clear, the added value of GSDM applied to ECCC forecasts was difficult to detect. 398
GSDM did not reduce the bias of RDPS and GEPS forecasts (Figure 3). The Rd product, 399
an ensemble at high-resolution (10 km) from the deterministic forecast RDPS, had 400
smaller CRPS values than Rn, an ensemble formed by the raw RDPS rainfall depth in the 401
neighboring pixels (Figure 5). However, consistency of Rd was not as strong compared to 402
that of Rn (Figures 6-8). Including the neighboring pixels in the ensemble forecast (Rdn) 403
increased the consistency. The added value of Rdn compared to Rn was the reduction of 404
the bias for convective events (Figure 8). For GEPS, GSDM did not improve the forecast, 405
20
except for the consistency of large stratiform rainfall depths, provided that the depths 406
generated in the neighboring pixels be included in the ensemble (Gdn, Figure 7). 407
To summarize, none of the analyzed products provided entirely satisfactory outputs for 408
short-term forecasts. That being said, there is potential for improvements. Integration of 409
a Bayesian approach in GSDM parameter estimation and data assimilation represent one 410
potential improvement. Instead of having the same parameter set for all realizations, the 411
parameter values could be selected from a predefined random distribution for each 412
realization. This would add some randomness to the disaggregated field, while keeping 413
spatial coherence. Similarly, the assumption that the low-resolution rainfall depth used as 414
input is reliable, which is realistic for reanalyses, but not necessarily true in a forecast 415
mode, could be relaxed. Random perturbations could be generated at each realization for 416
the mean areal rainfall depth used as input, as well as for wind speed, wind direction and 417
CAPE values. Finally, neighboring pixels could still be considered, but the number of 418
neighboring pixels could be reduced and/or vary depending on the type of event 419
(stratiform or convective). However, in-depth analyses on the extent of the neighboring 420
area were beyond the scope of this study. 421
From a wider perspective, ongoing improvements of meteorological modeling, including 422
parameterization, data assimilation, spatial resolution, and uncertainty estimation, is at 423
the heart of the matter. With better meteorological forecasts, new goals will become 424
obtainable and spatial disaggregation models or other statistical downscaling techniques 425
will remain of interest. Indeed, these techniques all need accurate input data. This work 426
focused on derived products for end users by coupling GSDM with currently available 427
21
meteorological products from ECCC. Meteorological model improvement was beyond 428
the scope of this study. 429
5 Conclusion
430A total of 30, 6-h, rainfall events within an area of about 40,000 km2 were analyzed to 431
evaluate accuracy, precision and consistency of the Gibbs Sampling Disaggregation 432
Model (GSDM, Gagnon, 2012; Gagnon and Rousseau, 2014) coupled with Environment 433
and Climate Change Canada (ECCC) meteorological short-term forecasts. The goal was 434
to produce reliable information at local scale (10 km) for end users. GSDM ran 435
sufficiently fast to provide an ensemble of rainfall fields for short-term forecasts. 436
Overall, GSDM applied with 100-km aggregated reference rainfall depths as input gave 437
accurate (low bias) and precise (low variability of the error and low dispersion of the 438
ensemble) 10-km fields. For small convective rainfall depths, GSDM was consistent 439
(observed value indistinguishable of a randomly selected realization of the ensemble), but 440
it could be improved for stratiform rainfall depths. When applied in forecast mode, 441
GSDM inherited the bias of the meteorological forecast. In the end, the 10-km 442
disaggregated rainfall depths from 100-km Global Ensemble Prediction System (GEPS) 443
forecasts, which were found to be highly biased and imprecise, resulted in biased and 444
imprecise information. 445
The Regional Deterministic Prediction System (RDPS) provided 10-km rainfall depths 446
with moderate biases. It is worth noting that by aggregating RDPS forecasts to a 100-km 447
spatial scale, biases evaluated on km pixels slightly increased compared to the raw 10-448
km forecasts, but remained much smaller than those from 100-km GEPS forecasts. The 449
22
GSDM coupled with 100-km aggregated RDPS forecasts produced better results than 450
with GEPS forecasts. However, despite this improvement, the disaggregated forecasts 451
were not consistent. Including the 120 neighboring pixels in the disaggregated ensemble 452
could help to mitigate the lack of consistency of the forecast, especially for convective 453
rainfall. 454
Possible areas for improvements were identified, such as a Bayesian estimation of GSDM 455
parameters, random perturbations of GSDM inputs and inclusion of a variable number of 456
neighboring pixels in the ensemble, where the exact number could depend on the type of 457
events. These improvements, once ascertained, would remain of interest even if 458
meteorological models were improved. 459
The same analyses for different experimental set ups could produce different outcomes. 460
In the present study, the area was mostly flat, except for the Appalachian Mountains in 461
the south-east portion of the study region. Application in a more complex topographical 462
region could require modifications to GSDM (Gagnon et al., 2013). Also, if one is 463
interested in 6-h rainfall depths at longer lead times (3, 10, 30 days), the meteorological 464
forecast bias and imprecision should increase, resulting in a decrease in the reliability of 465
the disaggregated rainfall depths. This effect could be reduced if one is interested in 466
cumulative rainfall depths instead of a specific 6-h period. Longer time steps would 467
smooth out the spatial variability and might increase the reliability of the forecasts, 468
disaggregated or not. 469
23
Acknowledgements
470
This project was funded by the Discovery Grant program of the Natural Sciences and 471
Engineering Research Council (NSERC) of Canada (Alain N. Rousseau, principal 472
investigator). The authors would like to thank the anonymous reviewers and the associate 473
editor for their helpful and constructive comments. The authors would also like to thank 474
Alexandre Vanasse from Solutions Mesonet for providing calibration data. 475
References
476Anderson, J.L. (1997). The impact of dynamical constraints on the selection of initial 477
conditions on ensemble predictions: Low-order perfect model results. Monthly Weather 478
Review, 125, 2969-2983. 479
Badas, M.G., R. Deidda, and E. Piga (2006). Modulation of homogeneous space-time 480
rainfall cascades to account for orographic influences, Natural Hazards and Earth System 481
Sciences, 6, 427-437, ISSN: 1561-8633. 482
Baldauf, M., Seifert, A., Förstner, J., Majewski, D., and Raschendorfer, M. (2011). 483
Operational convective-scale numeral weather prediction with the COSMO model: 484
Description and sensitivities. Monthly Weather Review, 139, 3887-3905. 485
Bauer, P., Thorpe, A., and Brunet, G. (2015). The quiet revolution of numerical weather 486
prediction. Nature, 525, 47-55. 487
Bendre, M.R., Thool, R.C., and Thool, V.R. (2015). Big data in precision agriculture: 488
Weather forecasting for future farming. 2015 1st International Conference on Next 489
24
Generation Computing Technologies (NGCT), Dehradun, pp. 744-750. doi: 490
10.1109/NGCT.2015.7375220 491
Cai, J., Liu, Y., Lei, T., and Pereira, LS. (2007). Estimating reference evapotranspiration 492
with the FAO Penman–Monteith equation using daily weather forecast messages. 493
Agricultural and Forest Meteorology, 145(1-2), 22-35. 494
Cai, X., Hejazi, M., and Wang, D. (2011). Value of Probabilistic Weather Forecasts: 495
Assessment by Real-Time Optimization of Irrigation Scheduling. Journal of Water 496
Resources Planning and Management, 137(5), 391-403. 497
Caron, J.-F., Milewski, T., Buehner, M., Fillion, L., Reszka, M., Macpherson, S., and St-498
James, J. (2015). Implementation of Deterministic Weather Forecasting Systems based 499
on Ensemble-Variational Data Assimilation at Environment Canada. Part II: The 500
Regional System. Monthly Weather Review, 143(7), 2560-2580. 501
Charron, M., Pellerin, G., Spacek, L., Houtekamer, P.L., Gagnon, N., Mitchell, H.L., and 502
Michelin, L. (2010). Toward Random Sampling of Model Error in the Canadian 503
Ensemble Prediction System. Monthly Weather Review, 138, 1877-1901. 504
Charron, M., Parent, A., and Frenette, R. (2013). The Regional Ensemble Prediction 505
System (REPS). RPN Seminar Series, Environment Canada, June 17th, 2013. 506
Deidda, R. (2000). Rainfall downscaling in a space-time multifractal framework, Water 507
Resources Research, 36(7), 1779-1794, ISSN: 0043-1397. 508
Deidda, R., M.G. Badas, and E. Piga (2006a). Space-time Multifractality of Remotely 509
Sensed Rainfall Fields, Journal of Hydrology, 322, 2-13. 510
25
Deidda, R., Badas, M.G., Seoni, A., and Piga, E. (2006b). A meteo-hydrological 511
forecasting chain: performance of the downscaling and rainfall-runoff steps in a small 512
catchment. Advances in Geosciences, 7, 361-369. 513
Fiorucci, P., La Barbera, P., Lanza, L.G., and Minciardi, R. (2001). A geostatistical 514
approach to multisensor rain field reconstruction and downscaling. Hydrology and Earth 515
System Sciences, 5(2), 201-213. 516
Forman, B.A., Vivoni, E.R., and Margulis, S.A. (2008). Evaluation of ensemble-based 517
distributed hydrologic model response with disaggregated precipitation products. Water 518
Resources Research, 44, W12409. 519
Fortin, V., G. Roy, N. Donaldson and A. Mahidjiba (2015). Assimilation of radar 520
quantitative precipitation estimations in the Canadian Precipitation Analysis (CaPA). 521
Journal of Hydrology, 531(2), 296–307. 522
Gaborit, E., Anctil, F., Fortin, V., and Pelletier, G. (2014). Hydrologic evaluation of 523
spatially disaggregated global ensemble rainfall forecasts. Hydrological Processes, 524
28(17), 4682-4693. 525
Gagnon, P. (2012). Désagrégation statistique de la précipitation mésoéchelle. Ph. D. 526
thesis. Institut National de la Recherche Scientifique, Centre eau, terre et environnement, 527
Université du Québec, Québec city, PQ, Canada, 245 pp. [in French] 528
Gagnon, P., Rousseau, A.N., Mailhot, A., and Caya, D. (2012). Spatial disaggregation of 529
mean areal rainfall using Gibbs sampling. Journal of Hydrometeorology, 13(1), 324-337. 530
26
Gagnon, P., Rousseau, A.N., Mailhot, A., and Caya, D. (2013). A Gibbs sampling 531
disaggregation model for orographic precipitation. Journal of Applied Earth Observation 532
and Geoinformation, 22(1), 16-26. 533
Gagnon, P., and Rousseau, A.N. (2014). Stochastic spatial disaggregation of extreme 534
precipitation to validate a Regional Climate Model and to evaluate climate change 535
impacts over a small watershed. Hydrology and Earth System Sciences, 18, 1695-1704. 536
Geman, S., and Geman, D. (1984). Stochastic relaxation, Gibbs distribution and the 537
Bayesian restoration of images. IEEE Transactions on Pattern Analysis Machine and 538
Intelligence, 6(6), 721-741. 539
Groppelli, B., Bocchiola, D., and Rosso, R. (2011). Spatial downscaling of precipitation 540
from GCMs for climate change projections using random cascades: A case study in Italy. 541
Water Resources Research, 47, W03519. 542
Gupta, V. K., and E. C. Waymire (1993), A statistical analysis of mesoscale rainfall as a 543
random cascade, Journal of Applied Meteorology, 32, 251-267. 544
Harris, D., and Foufoula-Georgiou, E. (2001). Subgrid variability and stochastic 545
downscaling of modeled clouds: Effects on radiative transfer computations for rainfall 546
retrieval, Journal of Geophysical Research, 106(D10), 10349-10362. 547
Hersbach, H. (2000). Decomposition of the Continuous Ranked Probability Score for 548
Ensemble Prediction Systems. Weather Forecasting, 15, 559-570. 549
27
Houtekamer, P.L., Deng, X., Mitchell, H.L., Baek, S.J., and Gagnon, N. (2014). Higher 550
Resolution in an Operational Ensemble Kalman Filter, Monthly Weather Review, 142, 551
1143-1162. 552
Hubert, P., Tessier, Y., Lovejoy, S., Schertzer, D., Schmitt, F., Ladoy, P., Carbonnel, J.P., 553
Violette, S., and Desurosne, I. (1993). Multifractals and extreme rainfall events, 554
Geophysical Research Letters, 20(10), 931-934. 555
Lovejoy, S., and Schertzer, D. (2010a). On the simulation of continuous in scale 556
universal multifractals, part I: Spatially continuous processes. Computers and 557
Geosciences, 36(11), 1393-1403. 558
Lovejoy, S., and Schertzer, D. (2010b). On the simulation of continuous in scale 559
universal multifractals, part II: Space-time processes and finite size corrections. 560
Computers and Geosciences, 36(11), 1404-1413. 561
Kumar, P., and Foufoula-Georgiou, E. (1993a). A multicomponent decomposition of 562
spatial rainfall fields, 1, Segregation of large- and small- scale features using wavelet 563
tranform, Water Resources Research, 29(8), 2515-2532. 564
Kumar, P., and Foufoula-Georgiou, E. (1993b). A multicomponent decomposition of 565
spatial rainfall fields, 2, Self-similarity in fluctuations, Water Resources Research, 29(8), 566
2533-2544. 567
Mailhot, J., Bélair, S., Charron, M., Doyle, C., Joe, P., Abrahamowicz, M., Bernier, N.B., 568
Denis, B., Erfani, A., Frenette, R., Giguère, A., Isaac, G.A., McLennan, N., McTaggart-569
Cowan, R., Milbrandt, J., and Tong, L. (2010). Environment Canada's Experimental 570
28
Numerical Weather Prediction Systems for the Vancouver 2010 Winter Olympic and 571
Paralympic Games. Bulletin of the American Meteorological Society, 91, 1073-1085. 572
Marsan, D., Schertzer, D., and Lovejoy, S. (1996). Casual space-time multi-fractal 573
processes: Predictability and forecasting of rain fields, Journal of Geophysical Research, 574
101(D21), 26,333-26,346. 575
Matheson, J.E., and Winkler, R.L. (1976). Scoring rules for continuous probability 576
distributions. Management Science, 22, 1087-1095. 577
Niemi, T.J., Guillaume, J.H.A., Kokkonen, T., Hoang, T.M.T., and Seed, A.W. (2016). 578
Role of spatial anisotropy in design storm generation: Experiment and interpretation, 579
Water Resources Research, 52, 69-89. 580
Olson, J., and Niemczynowicz, J. (1996). Multifractal analysis of daily spatial rainfall 581
distributions, Journal of Hydrology, 187, 29-43. 582
Over, T.M., and Gupta, V.K. (1996). A space-time theory of mesoscale rainfall using 583
random cascades. Journal of Geophysical Research, 101(D21), 26319-26331. 584
Paschalis, A., Molnar, P., Fatichi, S., and Burlando, P. (2013). A stochastic model for 585
high-resolution space-time precipitation simulation. Water Resources Research, 49, 586
8400-8417. 587
Peleg, N., and Morin, E. (2014). Stochastic convective rain-field simulation using a high-588
resolution synoptically conditioned weather generator (HiReS-WG). Water Resources 589
Research, 50(3), 2124-2139. 590
29
Perica, S., and Foufoula-Georgiou, E. (1996). Model for multiscale disaggregation of 591
spatial rainfall based on coupling meteorological and scaling descriptions. Journal of 592
Geophysical Research, 101(D21), 26347-26361. 593
Roberts, G.O. and Smith, A.F.M. (1994). Simple conditions for the convergence of the 594
Gibbs sampler and Metropolis-Hastings algorithms, Stochastic Processes and Their 595
Applications, 49, 207-216. 596
Silva, D., Meza, F.J., and Varas, E. (2010). Estimating reference evapotranspiration 597
(ETo) using numerical weather forecast data in Central Chile. Journal of Hydrology, 382, 598
64-71. 599
30
Tables
600
Table 1. Main characteristics of the 30 GEPS events analyzed. The values are the averages from the four 100-km grid cells analyzed 601
(2 x 2 central grid cells in Figure 2). 602 Date/Time (UTC) Rainfall depth (mm/6h) CAPE (J/kg) Wind at 700 hPa Date/Time (UTC) Rainfall depth (mm/6h) CAPE (J/kg) Wind at 700 hPa Speed (m/s) Direction (°) Speed (m/s) Direction (°) 2015-07-18 6:00 3.2 1 13.0 223 2015-09-14 6:00 3.2 16 5.9 198 2015-07-18 12:00 5.6 108 14.3 260 2015-09-14 12:00 6.1 5 7.9 100 2015-07-20 0:00 4.7 1475 13.8 273 2015-09-20 6:00 4.6 32 21.7 236 2015-07-26 6:00 4.9 45 12.4 270 2015-09-20 12:00 4.9 0 15.6 253 2015-07-26 12:00 10.0 79 10.3 248 2015-09-30 0:00 9.5 44 13.6 233 2015-07-30 18:00 7.1 593 13.8 238 2015-09-30 6:00 8.1 9 11.2 253 2015-08-04 0:00 4.1 392 15.0 224 2015-09-30 12:00 20.6 0 6.2 238 2015-09-03 12:00 4.2 568 8.4 293 2015-09-30 18:00 11.0 0 4.4 211 2015-09-09 0:00 3.6 305 10.5 239 2015-10-01 0:00 12.8 0 7.8 175 2015-09-10 0:00 4.6 161 13.6 254 2015-10-17 12:00 3.4 0 12.7 238 2015-09-11 12:00 4.1 2 4.9 240 2015-10-25 12:00 6.2 0 23.4 229 2015-09-13 0:00 3.8 56 9.8 223 2015-10-29 0:00 8.2 0 28.9 226 2015-09-13 6:00 8.9 24 10.0 193 2015-10-29 6:00 8.9 0 28.3 219 2015-09-13 12:00 11.4 7 8.3 192 2015-10-29 12:00 14.2 75 21.5 223 2015-09-14 0:00 3.3 154 9.2 197 2015-11-14 0:00 3.22 0 15.1 280 603 604
31
Table 2. Description of the rainfall series analyzed. Raw ECCC products appear in 605
boldface. For rainfall products aggregated before disaggregation (Cd, Cdn, Rd, Rdn), 606
wind and CAPE data were aggregated as well. 607
ID Source Resolution Nr of members of the ensemble
C CaPA 10 km 1 (deterministic, reference)
Cn CaPA 10 km 120 neighboring pixels
Ca CaPA aggregated 100 km 1 (deterministic)
Cd CaPA aggregated/disaggregated 10 km 100 random realizations Cdn CaPA aggregated/disaggregated 10 km 12,100 = 100 random realizations x
121 neighboring pixels
R RDPS 10 km 1 (deterministic)
Rn RDPS 10 km 121 neighboring pixels
Ra RDPS aggregated 100 km 1 (deterministic)
Rd RDPS aggregated/disaggregated 10 km 100 random realizations Rdn RDPS aggregated/disaggregated 10 km 12,100 = 121 neighboring pixels x
100 random realizations
G GEPS 100 km 21 members
Gd GEPS disaggregated 10 km 105 = 21 members x 5 realizations Gdn GEPS disaggregated 10 km 12,705 = 121 neighboring pixels x
21 members x 5 realizations 608
32
Table 3. Groups of pixels on which performance metrics were calculated. 610
Type of events Group ID CaPA rainfall depth
(mm) Number of pixels Stratiform (CAPE < 500 J/kg) 1 [0, 0.1) 2180 2 [0.1, 5) 4635 3 [5, 10) 2000 4 > 10 1885
Total stratiform 10,700 (111 tiles) Convective (CAPE > 500 J/kg) 5 [0, 0.1) 742 6 [0.1, 5) 409 7 > 5 149
Total convective 1,300 (13 tiles) Total 12,000 (120 tiles = 30 events x 4 tiles/event) 611 612
33
Figure Captions
613
Figure 1. Example of a 6-h rainfall event: depths from the 10-km reference data (left 614
panel), the aggregated 100-km reference data used as input by GSDM (middle panel) and 615
a realization of GSDM at 10-km (right panel). 616
Figure 2. Study area with the 40 x 40 10-km pixels (dotted lines) in the 4 x 4 100-km grid 617
cells (solid lines). The Bécancour watershed is shown for illustrative purposes. 618
Figure 3. Absolute value of the mean error (bias) (i.e. rainfall depth difference with the 619
reference depth C) of all products for the seven groups of pixels. IDs of the rainfall 620
products are defined in Table 2. IDs of the seven groups of pixels are defined in Table 3. 621
Figure 4. Standard deviation of the error of all products for the seven groups of pixels. 622
IDs of the rainfall products are defined in Table 2. IDs of the seven groups of pixels are 623
defined in Table 3. 624
Figure 5. Mean CRPS values of the ensemble products for the seven groups of pixels. IDs 625
of the rainfall products are defined in Table 2. IDs of the seven groups of pixels are 626
defined in Table 3. 627
Figure 6. Rank histograms of each ensemble product for group of pixels 2 (stratiform 628
rainfall depth between 0.1 and 5 mm; 4635 pixels). The dashed line illustrates a rank 629
histogram for a perfectly consistent ensemble product. 630
Figure 7. Rank histograms of each ensemble product for group of pixels 4 (stratiform 631
rainfall larger than 10 mm; 1885 pixels). The dashed line illustrates a rank histogram for 632
a perfectly consistent ensemble product. 633
34
Figure 8 Rank histograms of each ensemble product for group of pixels 6 (convective 634
rainfall depth between 0.1 and 5 mm; 409 pixels). The dashed line illustrates a rank 635
histogram for a perfectly consistent ensemble product. 636
35
Highlights
638
• GSDM applied on reference data generates accurate and precise rainfall fields 639
• GSDM inherits of the forecast bias 640
• Global ensemble forecasts (GEPS) are not consistent, with or without GSDM 641
• GSDM performance was better with regional deterministic forecasts (RDPS) 642
• Neighboring pixels should be considered when producing high-resolution 643
ensembles 644
645
